A330381 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - 5\,x - 10\,x^{2} - 9\,x^{3} - 3\,x^{4}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	1	7	9													17
3	1	12	34	27												74
4	1	18	83	142	81											325
5	1	25	165	451	547	243										1432
6	1	33	290	1117	2142	2005	729									6317
7	1	42	469	2373	6357	9341	7108	2187								27878
8	1	52	714	4536	15843	32328	38410	24604	6561							123049
9	1	63	1038	8022	34947	92497	151930	151314	83653	19683						543148
10	1	75	1455	13362	70389	231317	490925	673530	576959	280483	59049					2397545
11	1	88	1980	21219	132099	522786	1372856	2426165	2854653	2144218	930022	177147				10583234
12	1	102	2629	32406	234234	1091398	3440813	7503498	11343642	11675098	7805541	3055786	531441			46716589
13	1	117	3419	47905	396396	2136992	7909707	20657157	38449560	50738092	46384839	27933869	9964519	1594323		206216896
14	1	133	4368	68887	645073	3968250	16949673	51853011	115351173	187057906	218860285	179909391	98549092	32285041	4782969	910285253

Technical details

Node id: n-d697df4adfdac804 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-f2b7ac8f0752ccba

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  7,    9;
  3: 1, 12,   34,   27;
  4: 1, 18,   83,  142,    81;
  5: 1, 25,  165,  451,   547,   243;
  6: 1, 33,  290, 1117,  2142,  2005,    729;
  7: 1, 42,  469, 2373,  6357,  9341,   7108,   2187;
  8: 1, 52,  714, 4536, 15843, 32328,  38410,  24604,  6561;
  9: 1, 63, 1038, 8022, 34947, 92497, 151930, 151314, 83653, 19683;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(1+5*x+10*x^2+9*x^3+3*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A104029 → basis A007318

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003